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Lecture II - Preliminaries from General Topology

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Lecture II - Preliminaries from General Topology Lecture II - Preliminaries from general topology: We discuss in this lecture a few notions of general topology that are covered in earlier courses but are of frequent use in algebraic topology. We shall prove the existence of Lebesgue number for a covering, introduce the notion of proper maps and discuss in some detail the stereographic projection and Alexandroff's one point compactification. We shall also discuss an important example based on the fact that the sphere Sn is the one point compactifiaction of Rn. Let us begin by recalling the basic definition of compactness and the statement of the Heine Borel theorem. Definition 2.1: A space X is said to be compact if every open cover of X has a finite sub-cover. If X is a metric space, this is equivalent to the statement that every sequence has a convergent subsequence. If X is a topological space and A is a subset of X we say that A is compact if it is so as a topological space with the subspace topology. This is the same as saying that every covering of A by open sets in X admits a finite subcovering. It is clear that a closed subset of a compact subset is necessarily compact. However a compact set need not be closed as can be seen by looking at X endowed the indiscrete topology, where every subset of X is compact. However, if X is a Hausdorff space then compact subsets are necessarily closed. We shall always work with Hausdorff spaces in this course. For subsets of Rn we have the following powerful result. Theorem 2.1(Heine Borel): A subset of Rn is compact (with respect to the subspace topology) if and only if it is closed and bounded. The theorem provides a profusion of examples of compact spaces. 1. The unit sphere Sn−1 = (x ; x ; : : : ; x ) Rn x2 + x2 + + x2 = 1 is compact. f 1 2 n 2 j 1 2 · · · n g 2. The unit square I 2 = [0; 1] [0; 1] is compact. × 3. The set of all 3 3 matrices is clearly homeomorphic to R9. Then the set of all 3 3 orthogonal matrices, denoted× by O(3; R) is is compact. That is to say the orthogonal group is×compact. The result readily generalizes to the group of n n orthogonal matrices. × 4. Think of the set of all n n matrices with complex entries as Cn2 which in turn may be viewed as R2n2 . The set of all n× n unitary matrices is then easily seen to be a compact space. These × matrices form a group known as the unitary group U(n). 5. The set of all n n unitary matrices with determinant one is also a closed bounded subset of Cn2 and so is compact.× This is the special unitary group SU(n). Theorem 2.2: Suppose that X is a compact topological space, Y is an arbitrary Hausdorff space and f : X Y is a continuous surjection then −! 1. Y is compact. 2. If A is a closed subset of X then f(A) is closed. 3. It f is bijective then f is a homeomorphism. 9 Proof: The first assertion is proved in courses on point set topology. We remark that the Hausdorff assumption is not necessary for (i). The second follows from the first and we shall prove the third which will be of immense use in the sequel. Let g be the inverse of f and A be closed in X then g−1(A) = f(A) is closed in Y from which continuity of g follows. Definition 2.2 (The Lebsesgue number for a cover): Given an open covering Gα of a metric space X, a Lebesgue number for the covering is a positive number such that every ballf ofg radius is contained in some member Gα of the cover. Theorem 2.3: Every open covering of a compact metric space has a Lebesgue number. Proof: The student is advised to draw relevant pictures as he reads on. Suppose that a cover G f αg has no Lebesgue number. Then for every n N, 1=n is not a Lebesgue number and so there is a point 2 xn X such that the ball of radius 1=n centered at xn is not contained in any of the open sets in the2covering. By compactness the sequence x has a convergent subsequence converging to a point f ng p X. Choose an α such that G contains p and there is a δ > 0 such that the ball of radius δ around 2 α p is contained in Gα. Now take n large enough that 1=n < δ=3 and xn is contained in the ball of radius δ=3 centered at p. Now, since the ball of radius 1=n with center xn is not contained in any of the open sets in our covering, there exists y X such that y = G and d(x ; y ) < 1=n. But n 2 n 2 α n n d(p; y ) d(p; x ) + d(x ; y ) < 2δ=3 < δ: n ≤ n n n So y is in the ball of radius δ centered at p and so y G which is a contradiction. n n 2 α Definition 2.3 (Locally compact spaces): A (Hausdorff) space X is said to be locally compact if each point of X has a neighborhood whose closure is compact. It is an exercise for the student to check that under this hypothesis each point of X has a local base of consisting of compact neighborhoods. Examples: The reader may easily verify the following. 1. Open subsets of Rn are locally compact. 2. Q is not locally compact. Locally compact spaces are easily realized as dense open subsets of compact spaces. One has to merely adjoin one additional point. The idea is important in many applications and is called Alexandroff's one point compactification. One point compactification: Let X be a locally compact, non-compact Hausdorff space and X = X be the one point union of X with an additional point . The topology consists of all the op[enf1gsubsets in X as well as all the subsets of the form 1(X K), where KT ranges over f1g [ − allb the compact subsets of X. The following theorem summarizes the properties of X and the proof is left for the reader. b 10 Theorem 2.4: (i) The collection of sets is a topology on X. T (ii) The family of sets X K, where K ranges over all compact subsets of X, forms a neighborhood − base of . b 1 (iii) X with the given topb ology is an open dense subset of X. (iv) The space X is compact. b Definition 2.4 (Propb er maps): A map f : X Y between topological spaces is said to be proper if f −1(C) is a compact subset of X whenever−C!is a compact subset of Y . Theorem 2.5: Suppose X and Y are locally compact spaces and f : X Y is a continuous proper −! map then it extends continuously as a map f : X Y between their one point compactifications. −! Proof: Denote the adjoined points in X andb bY as pband q respectively and extend the given map by sending p to q. We need to show that the extension is continuous at p. Let C be any compact subset of Y so that K = f −1(C) is compactb in bX. Then N = X K is a neighborhood of p in X − that is mapped by f into the preassigned neighborhood Y C of q. This proves the continuity of the extension. − b b The converse is bnot true as the constant map shows. bHowever the following version in the reverse direction is easy to see, Theorem 2.6: Suppose X and Y are locally compact Hausdorff spaces and f : X Y is a continuous map such that f −1(q) = p , where p and q are as in the previous theorem,−!then the f g restriction of f to X is a proper map. b b Proof: If C is a compact subset of Y then f −1(C) being a closed subset of X is compact. The hypothesis says that f −1(C) does not contain p and hence is a compact subset of X itself. b Stereographic projection: Consider the sphere Sn = (x ; x ; : : : ; x ) Rn+1 x2 + x2 + + x2 = 1 1 2 n+1 2 j 1 2 · · · n+1 n o and the plane xn+1 = 0 of the equator. Let n = (0; 0; : : : ; 0; 1) and x be a general point on the equatorial plane. The line through n and x is described parametrically by (1 t)n + tx and meets the sphere at points corresponding to the roots of the quadratic equation − (1 t)n + tx; (1 t)n + tx = 1: h − − i The root t = 0 corresponds to the point n and the second root 2(1 n x) t = − · 1 + x 2 2n x k k − · is continuous with respect to x and provides a point F (x) Sn n . The map F is a bijective n 2 − f g continuous map between the plane xn+1 = 0 and S n . Note that the origin is mapped to the south pole by F . The inverse map G is called the stereographic− f g projection. Let us now show that G is also continuous whereby it follows that F is a homeomorphism. 11 Well, let y be a point on the sphere minus the north pole n.
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